Millersville University Department of Mathematics MATH 211, Calculus II, Test 1 February 8, 2011
Please answer the following questions. Your answers will be evaluated on their correctness, completeness, and use of mathematical concepts we
The Ratio Test and the Root Test
It is a fact that an increasing sequence of real numbers that is bounded above must converge.
The picture shows an increasing sequence that is bounded above by some number M . It seems reasonable
that the terms
Partial fractions is the opposite of adding fractions over a common denominator. It applies to integrals
of the form
dx, whereP (x)
Q(x) are polynomials.
into a sum of smaller terms which are easier to inte
When an integral contains a quadratic expression ax2 + bx + c, you can sometimes simplify the integrand
by completing the square. This eliminates the middle term of the quadratic; the resulting integral can
then be co
Log and Exponential Derivatives
Here are the formulas for the derivatives of ln x and ex :
ln x =
e = ex .
Ill derive them at the end. First, Ill give some examples to show how theyre used.
x ln x.
Parametric Equations of Curves
A pair of equations
x = f (t),
y = g(t),
are parametric equations for a curve. You graph the curve by plugging values of t into x and y, then
plotting the points as usual.
Example. The parametric equations
x = c
Absolute Maxima and Minima
Ill begin with a couple of examples to illustrate the kinds of problems I want to solve.
Example. A string 6 light years in length is cut into two pieces. One piece is used to make a circle, while
the other piece is u
The Mean Value Theorem
A secant line is a line drawn through two points on a curve.
The Mean Value Theorem relates the slope of a secant line to the slope of a tangent line.
The Mean Value Theorem. If f is continuous on a x b and dierentiable on
The Natural Logarithm
The Power Rule says
xn dx =
xn+1 + C
provided that n = 1. The formula does not apply to
An antiderivative F (x) of
would have to satisfy
F (x) = .
But the Fundamental Theorem implies that if x > 0
Properties of Limits
There are many rules for computing limits. Ill list the most important ones. There are analogous
results for left and right-hand limits; just replace lim with lim+ or lim .
I mentioned the rst rule earlier:
If f (
Left and Right-Hand Limits
In some cases, you let x approach the number a from the left or the right, rather than both sides at
once as usual.
lim f (x) means: Compute the limit of f (x) as x approaches a from the right.
lim f (x) means: C
Limits: An Introduction
Calculus was used long before it was established on rm mathematical foundations. Limits provide a
precise way of talking about convergence and innite processes.
For example, derivatives and integrals are dened using limit
Inverse Trig Functions
If you restrict f(x) = sin x to the interval
x , the function increases:
y = sin x
This implies that the function is one-to-one, and hence it has an inverse. The inverse is called the
inverse sine or arcsine
The Limit Denition
Having discussed how you can compute limits, I want to examine the denition of a limit in more detail.
1 cos(x8 )
. You might
think of drawing a graph; many graphing calculators, for instance, produce a graph like the o
Polar coordinates are an alternative to rectangular coordinates for referring to points in the plane.
A point in the plane has polar coordinates (r, ). r is (roughly) the distance from the origin to the point;
is the angle betw
Volumes of Revolution by Slicing
Start with an area a planar region which you can imagine as a piece of cardboard. The cardboard
is attached by one edge to a stick (the axis of revolution). As you spin the stick, the area revolves and
The work required to raise a weight of P pounds a distance of y feet is P y foot-pounds. (In m-k-s
units, one would say that a force of k newtons exerted over a distance of y feet does k y newton-meters, or
joules, of work.)
Example. If a 1
The Remainder Term and Error Estimation
If the Taylor series for a function f (x) is truncated at the nth term, what is the dierence between f (x)
and the value given by the nth Taylor polynomial? That is, what is the error involved in using the
You can use substitution to convert a complicated integral into a simpler one. In these problems, Ill
let u equal some convenient x-stu say u = f(x). To complete the substitution, I must also substitute
for dx. To do this, co
For trig integrals involving powers of sines and cosines, there are two important cases:
1. The integral contains an odd power of sine or cosine.
2. The integral contains only even powers of sines and cosines.
I will look
Constructing Taylor Series
The Taylor series for f(x) at x = c is
f(c) + f (c)(x c) +
f (n) (c)
(x c)2 +
(x c)3 + =
(x c)n .
(By convention, f (0) = f.) When c = 0, the series is called a Maclaurin series.
You can constru
Limits and Derivatives of Trig Functions
If you graph y = sin x and y = x, you see that the graphs become almost indistinguishable near x = 0:
That is, as x 0, x sin x. This approximation is often used in appl
Trig substitution reduces certain integrals to integrals of trig functions. The idea is to match the
given integral against one of the following trig identities:
1 (sin )2 = (cos )2
1 + (tan )2 = (sec )2
(sec )2 1 = (tan )2
Tangent Lines and Rates of Change
Given a function y = f (x), how do you nd the slope of the tangent line to the graph at the point
P (a, f (a)?
(Im thinking of the tangent line as a line that just skims the graph at (a, f (a), without going thr
An innite sequence is a list of numbers. The following examples should make the idea clear.
Example. Here is a familiar sequence:
1, 2, 4, 8, 16, . . . , 2n , . . .
Sequences are often written using subscript notation. This one might b
Related rates problems deal with situations in which several things are changing at rates which are
related. The way in which the rates are related often arises from geometry, for example.
Example. The radius of a circle increases
Review: Convergence Tests for Innite Series
When you are testing a series for convergence or divergence, its helpful to run through your list of
convergence tests if you dont see what to do immediately just as you might run through your list of
Summation notation is used to denote a sum of terms. Usually, the terms follow a pattern or formula.
is shorthand for f(0) + f(1) + + f(n).
In this case, 1 and n are the limits of summation and k is the summation v
LHpitals Rule is a method for computing a limit of the form
c can be a number, +, or . The conditions for applying it are:
1. The functions f and g are dierentiable in an open interval containing c. (c may al